Page 207 - Advances in Biomechanics and Tissue Regeneration

202 10. DETERMINATION OF THE ANISOTROPIC MECHANICAL PROPERTIES OF BONE TISSUE

mechanoregulatory with the bioregulatory. The first mathematical model related to mechanoregulation of bone remo-
deling that described Wolff’s law was developed in 1960 by Pauwels [20]; it was later applied in 1965 [22]. Wolff’s law,
developed by anatomist and surgeon Julius Wolff, states that bone adapts itself to applied loads. Wolff reported that
the directions of the applied external loads directly influence the direction of the trabecular bone by changing the tra-
becular bone’s physical disposition and distribution. Today, it is generally accepted that bone remodeling is mainly
caused by the transient nature of its strain/stress fields, induced by the external loads applied in its physical boundary.
From 1960 until now, many other models were created by using novel ideas or by modifying/enhancing existent
models [23–39]. In 2012, Belinha et al. [40] developed a material law that permits correlating the bone apparent density
with the bone level of stress. Using this new material law, a biomechanical remodeling model was developed as an
adaptation of Carter’s models for predicting bone density distribution that assumes that bone structure is a gradually
self-optimizing anisotropic biological material that maximizes its own structural stiffness [27, 29, 30, 41]. Peyroteo et al.
[42, 43] developed another model, considering the Belinha et al. [40] material law as part of a mechanoregulation
model. Among many bioregulatory models [21, 44–48], one of the most known bioregulatory models is Komarova’s
model [45]. Komarova’s model describes the population dynamics of bone cells accordingly with the number of oste-
oclasts and osteoblasts at a single basic multicellular unit (BMU). The development of these mechanoregulatory and
bioregulatory models led to the development of mechanobioregulatory modes, being one of the first versions devel-
oped by Lacroix and coworkers in 2002. Using the models developed by Prendergast et al. [34] and Huiskes et al. [49],
this first model considered cellular processes together with mechanical factors by incorporating the random walk of
mesenchymal stem cells [35, 36]. One other numerical model was developed by Mousavi and Doweidar [50] that
allows studying mesenchymal stem cell differentiation to osteoblasts as well as osteoblast proliferation due to mechan-
ical stimulations. The latest developed mechanobiological remodeling model was developed in 2016, where it was
included in the hormonal regulation and biochemical coupling of bone cell populations, the mechanical adaptation
of the tissue, and factors that influence the microstructure on bone turnover rate [51].
One key factor in all these models is the characterization of the bone mechanical properties. The first models con-
sidered bone as an isotropic material, a simplistic approach to the behavior of trabecular bone that disregards the
importance of orientation in the remodeling process [31, 41, 52, 53]. Later, models started to consider material density
and orientation with bone anisotropic mechanical properties, taking into account the trabecular architecture features
[38, 54–56]. More recently, some works have started to characterize bone mechanical properties using the fabric tensor
concept [57–62]. The fabric tensor is a symmetric second-rank tensor that characterizes the arrangement of a multi-
phase material, encoding the orientation and anisotropy of the material.
Numerical methods combined with computer science are widely used in a variety of areas, from civil engineering to
mechanical engineering to chemistry to biomechanics. These methodologies allow studying and analyzing, in silicio,
the behavior of materials and structures, being first used in the biomechanics area in 1972 by Huiskes and coworkers
[63] to evaluate stresses in human bones. Today, FEM is one of the most popular discrete numerical methods [64] while
other methods such as meshless have started to appear. Meshless methods evolved from the first developed meshless
method dated from 1977, where Gingold et al. [65] proposed smoothed-particle hydrodynamics, to more recent
methods such as the natural neighbor radial point interpolation method (NNRPIM) [66] and the natural radial element
method (NREM) [67].
The main difference between meshless methods and the FEM is the methodology to discretize the problem domain.
Meshless methods, in opposition to FEM, do not use elements to establish nodal/element connectivity, and so discre-
tize the problem domain using an unstructured node set that can be distributed regularly or irregularly. Because of
this, meshless methods can have advantages such us the capability to discretize high complex problem domains using
information gathered directly from medical images, a feature of high importance in the biomechanics field.
In meshless methods, the nodal discretization is constructed just by using the spatial coordinates of the nodes,
allowing us to define individually the material properties of each node. The concept of influence domain, equivalent
to elements in FEM, defines how each node interacts with its neighbor by using geometrical and mathematical con-
structions. Only meshless methods that use nodal-dependent constructions of the integration mesh are called truly
meshless methods because they allow directly defining the spatial position and the integration weight of all integration
points only using the spatial positions of the nodes. The untrue meshless methods use a nodal-independent back-
ground integration mesh to establish the system of equations from the integro-differential equations ruling the phys-
ical phenomenon under study [1]. Meshless methods are used in many different fields of mechanics, such us laminates
[68–78], two-dimensional (2D) and three-dimensional (3D) linear elasticity [79], plate and shell bending problems [80],
composites [81, 82], fractures [83–85], etc. Meshless is also widely used in biomechanics. They are used, for example, to
study the behavior of bone response to the insertion of implants [86–89], to analyze the behavior of soft tissue under
stress [90, 91], to evaluate the behavior of the inner ear [92, 93], and for bone remodeling computational research [42, 43,
86, 87, 94–102].

I. BIOMECHANICS